List the critical points of the function for which the Second Partial Test fails.
A) Test fails at and .
B) Test fails only at .
C) Test fails at .
D) Test fails at .
E) Test fails only at .

Question

List the critical points of the function  for which the Second Partial Test fails.
A) Test fails at and . 
B) Test fails only at . 
C) Test fails at . 
D) Test fails at . 
E) Test fails only at .

Quizplus · Accepted Answer

The Second Partial Test is also known as the determinant test, which involves calculating the determinant of the Hessian matrix (matrix of second-order partial derivatives) at each critical point. If the determinant is positive, then the critical point is a local minimum. If the determinant is negative, then the critical point is a local maximum. If the determinant is zero, then the test fails and more information is needed.

In this question, we need to identify the critical points where the Second Partial Test fails, meaning that the determinant of the Hessian matrix is zero. Only option C lists such a critical point, 11eaa595_62d3_f6c4_a696_65b484345d13_TB4584_11_TB4584_11. Therefore, the answer is C.

List the Critical Points of the Function for Which